Space sustainability

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Space_sustainability 

 

A computer-generated animation by the European Space Agency representing space debris in low earth orbit at the current rate of growth compared to mitigation measures being taken.

Space sustainability aims to maintain the safety and health of the space environment.

Similar to sustainability initiatives on Earth, space sustainability seeks to use the environment of space to meet the current needs of society without compromising the needs of future generations. It usually focuses on space closest to Earth, Low Earth Orbit (LEO), since this environment is the one most used and therefore most relevant to humans. It also considers Geostationary Equatorial Orbit (GEO) as this orbit is another popular choice for Earth-orbiting mission designs.

The issue of space sustainability is a new phenomenon that is gaining more attention in recent years as the launching of satellites and other space objects has increased. These launches have resulted in more space debris orbiting Earth, hindering the ability of nations to operate in the space environment while increasing the risk of a future launch-related accident that could disrupt its proper use. Space weather also acts as an outstanding factor for spacecraft failure. The current protocol for spacecraft disposal at end-of-life has, at large, not been followed in mission designs and demands extraneous amounts of time for disposal.

Precedent created through prior policy initiatives has facilitated initial mitigation of space pollution and created a foundation for space sustainability efforts. To further mitigation, international and transdisciplinary consortia have stepped forward to analyze existing operations, develop standards, and incentivize future procedures to prioritize a sustainable approach. A shift towards sustainable interactions with the space environment is growing in urgency due to the implications of climate change and increasing risk to spacecraft as time presses on.

Fundamentals

Space sustainability requires all space participants to have three consensuses. The space field should be used peacefully, jointly protect the space field from harm, and maximize space utilization through environmental, economic, and security exploration of space. These consensuses also clarify the relationship between space sustainability and international security, that states and individuals explore space for various purposes. Their reliance on space needs to be guided by rules, order, and policies and obtain more benefits without negatively affecting the space environment and space activities.

However, striking an agreement remains challenging even with such demands in place. In the discussions between countries on long-term sustainability, technical improvements are given more importance than introducing and applying new legal regimes. Specifically, technical approaches to space debris have been proposed, such as debris removal. Specific data on space debris is also being explored to help study its impact on sustainability and promote further cooperation between countries.

Current state

Space sustainability comes into play to address the pressing current state of near-Earth orbits and its high amounts of orbital debris. Spacecraft collisions with orbital debris, space weather, overcrowding in low earth orbit (LEO) makes spacecraft susceptible to higher rates of failure. The current end-of-life protocol for spacecraft exacerbates the space sustainability crisis; many spacecraft are not properly disposed, which increasing the likelihood of further collisions.

Orbital debris

Orbital debris is defined as unmanned, inoperate objects that exist in space. This orbital debris breaks down further as time progresses as a result of naturally occurring events, such as high-velocity collisions with micrometeoroids, and forced events, such as a controlled release of a launch vehicle. In LEO, these collisions can take place at speeds anywhere between an average velocity of 9 kilometers per second (km/s) and 14 km/s relative to the debris and spacecraft. In GEO, however, these high-speed collisions are a much lower risk as the average relative velocity between the debris and spacecraft is typically between 0 km/s and 2.5 km/s. As of 2012, the United States Joint Space Operations Center tracked 21,000 pieces of orbital debris larger than 10 cm in Earth’s nearby orbits (LEO, GEO, and sun-synchronous), where 16,000 of these pieces are catalogued. Space debris can be categorized into three categories: small, medium, and large. Small debris is for pieces that are less than 10 centimeters (cm). Medium-sized debris is for pieces larger than 10 cm, but not an entire spacecraft. Large-sized debris has no official classification, but typically refers to entire spacecraft, such as an out of use satellite or launch vehicle. It is difficult to track small-sized debris in LEO, and challenging to track small and medium-sized debris in GEO. Yet this statement is not to discount the abilities of LEO and GEO tracking capabilities, the smallest piece of tracked debris can weigh as low as ten grams. If the size of the debris prohibits it from being tracked, it also cannot be avoided by the spacecraft and does not allow the spacecraft to lower its risk of collisions. The likelihood of the Kessler syndrome, which essentially states that each collision produces more debris, grows larger as the amount of orbital debris multiplies, increasing the amount of further collisions until space cannot be used entirely.

Space weather

Space weather poses a risk to satellite health, consequently, resulting in greater amounts of orbital debris. Space weather impacts satellite health in a variety of ways. Firstly, surface charging from the sun’s surface facilitates electrical discharges, damaging on-orbit electronics, posing a threat to mission failure. Single Event Upsets (SEUs) can also damage electronics. Dielectric charging and bulk charging can also occur, causing energy problems within the spacecraft. Additionally, at altitudes less than one thousand kilometers, atmospheric drag can increase during solar storms by increasing the altitude of the spacecraft, only adding more drag onto the spacecraft. These factors degrade performance over the spacecraft’s lifetime, leaving the spacecraft more susceptible to further system and mission failures.

Overcrowding

There has been a dramatic increase in the use of LEO and GEO orbits over the last sixty years since the first satellite launch in 1957. To date, there have been approximately ten thousand satellite launches, whereas only approximately 2000 are still active. These satellites can be used for a variety of purposes, which are telecommunications, navigation, weather monitoring, and exploration. Within the coming decade, companies like SpaceX are predicted to launch an additional fifteen thousand satellites into LEO and GEO orbits. Microsatellites built by universities or research organizations have also increased in popularity, contributing to the overcrowding of near earth orbits. This overcrowding of LEO and GEO orbits increases the likelihood of potential collisions among satellites and orbital debris, contributing further to the large amount of orbital debris present in space.

End of life protocol

The current end of life protocol is that at the end of mission, spacecraft are either added to the graveyard orbit or at a low enough altitude that drag will allow the spacecraft to burn up upon reentry and fall back to Earth. Approximately twenty satellites are put into the graveyard orbit each year, but this rate is expected to increase to one thousand satellites per year by 2050. There is no current process to return satellites to Earth after entering the graveyard orbit. The process of a spacecraft returning to Earth via drag can take between ten to one hundred years. This protocol is critical to reduce overcrowding in near-Earth orbits.

Mega constellation and space debris

The impact of constellations on the space environment has also been studied, such as the probability of collisions of mega constellations in the presence of large amounts of space debris. Although studies have shown that the predictors of mega constellations are highly variable, specific information related to mega constellations is not transparent.

But any catastrophic collision, as in the case of Kessler syndrome, has consequences for people and the environment. Putting this thinking into mega constellations, mega constellations existence may have potential benefits, but it will not bring adequate help to the governance of space debris. At the same time, the space debris situation cannot be underestimated or ignored because of the existence of mega constellations.

Concern

The existence of orbital debris has caused great trouble to the conduct of space activities. The development of space sustainability has not given sufficient political attention, although some warnings and discussions have made this abundantly clear. Debris management is still voluntary on the part of the state, and there are no laws mandating debris management practices, including the amount of debris to be managed. Although the UN Space Debris Mitigation Guidelines were promulgated in 2007 as an initial measure of space debris governance, there is still no broad consensus or action on further limits on space debris after that.

The difficulties for individuals wishing to participate in debris management initiatives cannot be ignored. Any individual or sector desiring to participate in space debris operations needs to obtain permission from the launching state, which is difficult for the launching state to do. This is because the process of space debris management inevitably has a negative impact on other space objects, and there is a lot of subsequent liability in terms of financial consumption. Therefore, the launching state would argue that space debris management requires the joint efforts of all states. However, it is difficult to determine what actions can be taken to gain acceptance between countries.

Regulations

Current space sustainability efforts rely heavily on the precedent set by regulatory agreements and conventions of the twentieth century. Much of this precedent is included in or is related to the Outer Space Treaty of 1963, which represented one of the initial major efforts by the United Nations to create legal frameworks for the operation of nations in space.

Pre-Outer Space Treaty

The international community has had concerns about space contamination since the 1950’s prior to the launch of Sputnik I. These concerns stemmed from the idea that increasing rates of exploration into further areas of outer space could lead to contamination capable of damaging other planetary bodies, resulting in limitations to human exploration on these bodies and potential harm to the Earth. Efforts to combat these concerns began in 1956 with the International Astronautical Federation (IAF) and the United Nations Committee on the Peaceful Uses of Outer Space (COPUOUS). These efforts continued to 1957 through the National Academy of Sciences and International Council for Science (ICSU). Each of these organizations aimed to study space contamination and develop strategies for how to best address its potential consequences. The ICSU went on to create the Committee on Contamination by Extraterrestrial Exploration (CETEX) that put forward recommendations leading to the establishment of the Committee on Space Research (COSPAR). COSPAR continues to address outer space research on an international scale today [cite cospar].

Outer Space Treaty

Relevant regulations of international space law to sustainability in space can be found in the Outer Space Treaty, which was adopted by the UN General Assembly in 1963. The Outer Space Treaty contains seventeen articles designed to create a basic framework for how international law can be applied in outer space. Basic principles of the Outer Space Treaty include the provision in Article IX that parties should "avoid harmful contamination of space and celestial bodies;"  definitions of "harmful contamination" are not provided. Other articles of relevance to space sustainability include articles I, II, and III that concern the fair and inclusive international use of space in a manner free from sovereignty, ownership, or occupation by any nation. In addition, articles VII and VIII protect ownership by their respective countries of any objects launched to space while attributing responsibility for any damages to the property or personnel of other countries by those objects to said countries. Descriptions or definitions for what these damages may entail are not provided.

COSPAR Planetary Protection Policy

Principles of Article IX provided the basis for the Committee of Space Research (COSPAR) Planetary Protection Policy guidelines, which are generally well-regarded among scientific experts. Such guidelines, however, are non-binding and often described as "soft-law," as they lack legal mandate. The Planetary Protection Policy is primarily concerned with providing information regarding best practices to avoid contamination of the space environment during space exploration missions. COSPAR believes that the prevention of such contamination is in the best interest of humanity as it may impede scientific progress, exploration, and the mission of a search for life. In addition, the argument is made that cross-contamination of the Earth can be potentially harmful to its environment due to the largely unknown nature of potential space contaminants.

Other relevant regulations

Regulatory clarifications concerning the Outer Space Treaty of 1963 of relevance to space sustainability were made in subsequent years. The 1967 Return Agreement relates mainly to the return of lost astronauts to their appropriate nations, but also requires Outer Space Treaty signing nations to assist other nations with the return of objects that return to Earth from orbit to their proper owners The 1972 Liability Convention attributes liability for damages from space objects to the nation that launched the object, regardless of whether that damage occurred in space or on Earth. Other clarifications include the 1975 registration convention that attempted to create mechanisms for nations to identify space objects, and the 1979 Moon Agreement that established protections for the environments of the Moon and other nearby planetary bodies. These agreements and conventions represented attempts to improve the initial Outer Space Treaty as space exploration continued to grow in importance throughout the 20th century.

Attitudes

Countries and major international institutions

Both the state and space agencies are working to improve the laws and regulations that facilitate the long-term sustainability of space. For example, the European Code of Conduct for Space Debris Mitigation signed by France, the UK and other countries in 2016. China, Brazil, Mexico and others have legal background and methodological measures under long-term space sustainability. However, the main problem is that until the concept of space sustainability is agreed between countries, inter-regional efforts are not working well.

Currently, the Committee on the Peaceful Uses of Outer Space (COPUOS) encourages states to incorporate the space debris mitigation guidelines developed by bodies such as the Inter-Agency Space Debris Coordination (IADC) into their national legislation, thereby regulating state behavior. Some countries have responded positively to this, such as Switzerland, the Netherlands and Spain. However, there are still some countries that do not consider debris management approaches in their national legislation, such as Japan and Australia. Many delegates at the COPUOS meeting expressed their reasons for doing so, arguing that space debris management is closely linked to technology and funding. Technology is dynamic and constantly evolving. Therefore, the incorporation of debris governance guidelines into national law is not an immediate priority at this time.

Scientific attitudes

A study outlined rationale for governance that regulates the current free externalization of true costs and risks, treating orbital space around the Earth as an "additional ecosystem" or a common "part of the human environment" which should be subject to the same concerns and regulations like e.g. oceans on Earth. While scientists may not have the means to make and enforce global laws themselves, the study concluded in 2022 that it needs "new policies, rules and regulations at national and international level".

Mitigation

Sustainability mitigation efforts include but are not limited to design specifications, policy change, removal of space debris, and restoration of orbiting semi-functional technologies. Efforts begin by regulating the debris released during normal operations and post-mission breakups. Due to the increased awareness of high-velocity collisions and orbital debris in the previous decades, missions have adapted design specifications to account for these risks. For example, the RADARSAT program implemented 17 kilograms of shielding to their spacecraft, which increased the program’s predicted success rate to 87% from 50%. Another effort in mitigation is restoring semi-functional satellites, which allows a spacecraft classified as “debris” to “functional.”  Space debris mitigation focuses on limiting debris release during normal operations, collisions and intentional destruction. Mitigation also includes reducing the possibility for post-mission breakups due to stored energy and/or operations phases, as well as addressing procedure for end of mission disposal for spacecraft.

Space Sustainability Rating

One example leading the regulatory sustainability measures is the Space Sustainability Rating (SSR), which is an instigator for industry competitors to incorporate sustainability into spacecraft design. The Space Sustainability Rating was first conceptualized at the World Economic Forum Global Future Council on Space Technologies designed by international and transdisciplinary consortia. The four leading organizations are the European Space Agency, Massachusetts Institute of Technology, University of Texas at Austin, and BryceTech with the goal to define the technical and programmatic aspects of the SSR. The SSR represents an innovative approach to combating orbital debris through incentivizing the industry to prioritize sustainable and responsible operations. This response entails the consideration of potential harm to the space environment and other spacecraft, all while maintaining mission objectives and high-quality service. The rating takes inspiration from other standards, like leadership in energy and environmental design (LEED) for the building sector. Several of the factors emphasized in the rating were extracted from LEED design considerations like the incorporation of feedback and public comments, or the rating’s advocacy to influence policy, such as orbit fragmentation risks, collision avoidance capabilities, trackability, and adoption of international standards.

Tracking

Tracking is one of the main Space Sustainability Rating modules’ efforts. The module "Detectability, Identification and Tracking" (DIT) consists of standardizing the comparison of satellite missions to encourage satellite operators to improve their satellite design and operational approaches for the observer to detect, identify, and track the satellites. Tracking presents challenges when the observer seeks to monitor and predict the spacecraft behavior over time. While the observer may know the name, owner, and instantaneous location of the satellite, the operator controls the full knowledge of the orbital parameters. The Space Situational Awareness (SSA) is one the tools geared towards solving the challenges presented when tracking orbiting satellites and debris. The SSA continuously tracks objects using ground-based radar and optical stations so the orbital paths of debris can be predicted and operations avoid collisions. It feeds data to 30 different systems like satellites, optical telescopes, radar systems, and supercomputers to predict risk of collision days in advance. Other efforts in tracking orbital debris are made by the US Space Surveillance Network (SSN).

Removal

Under the "External Services" module of the SSR, the rating offers commitment to use or demonstration of use of end-of-life removal services. Space debris mitigation measures are found to be inadequate to stabilize debris environments with an actual current compliance of approximately sixty percent. Moreover, a low compliance rate of approximately thirty percent of the 103 spacecraft that reached end of life between 1997 and 2003 were disposed of in a graveyard orbit. Since policy has not caught up to ensure the longevity of LEO for future generations, actions like Active Debris Removal (ADR) are being considered to stabilize the future of LEO environment. Most famous removal concepts are based on directed energy, momentum exchange or electrodynamics, aerodynamic drag augmentation, solar sails, auxiliary propulsion units, retarding surfaces and on-orbit capture. As ADR consists of an external disposal method to remove obsolete satellites or spacecraft fragments. Since large-sized debris objects in orbit provide a potential source for tens of thousands of fragments in the future, ADR efforts focus on objects with high mass and large cross-sectional areas, in densely populated regions, and at high altitudes; in this instance, retired satellites and rocket bodies are a priority. Other practical advancements toward space debris removal include missions like RemoveDEBRIS and End-of-Life Service (ELS-d).

Growing urgency

The growth of all tracked objects in space over time

The previous reduced state of regulation and mitigation on space debris and rocket fuel emissions is aggravating the Earth’s stratosphere through collisions and ozone depletion, increasing the risk for spacecraft health through its lifetime.

Inaccessibility to LEO

Due to the increase of satellites being launched and the growing amount of orbital debris in LEO, the risk of LEO becoming inaccessible over time (in accordance with the Kessler syndrome) is increasing in likelihood. The mitigation policies for creating space debris fall under an area of voluntary codes by the states, although it has been disputed whether the Article I Outer Space Treaty or the Article IX Outer Space Treaty protects the space environment from deliberate harm, which has yet to be upheld. In 2007, an inactive Chinese satellite was purposefully destroyed by the Chinese government as a part of their anti-satellite weapon test (ASAT), spreading nearly 2800 objects of space debris five centimeters or larger into LEO. An analysis concluded that about eighty percent of the debris will remain in LEO nine years after this destruction. In addition, the destruction increased the collision likelihood for three Italian satellites that launched the same year as the Fengyun-1C destruction. The increase in collision ranged between ten and sixty percent. However, there were no legal consequences against the Chinese government.

Rocket fuel emissions

When rockets are launched into space, parts of their fuel enter the stratosphere of the Earth. Rocket fuel emissions are made up of carbon dioxide, water, hydrochloric acid, alumina and soot particles. The most concerning emissions from rocket fuel are chlorine and alumina particles from solid rocket motors (SRMs) and soot from kerosene fueled engines. When the hydrochloric acid from the engine exhaust dissociates, the free chlorine roams freely in the stratosphere. The chemical reaction between these chlorine and alumina causes ozone depletion. In addition, the soot particles form over a black umbrella over the stratosphere which can cause the temperature of the Earth’s surface to lower and further depleting the ozone layer, an unintentional form of geoengineering. The nature of geoengineering has been disputed as a form of mitigating global warming and has the possibility of being banned and holding rockets accountable for the soot particles they distribute to the stratosphere. New types of engines and fuels are emerging, mainly the liquid oxygen (LOX) and monomethylhydrazine engine, but there is minimal research on their impact on the environment besides their emission of hydroxide and nitrogen oxide compounds, two molecules that have significant impact on the ozone layer. Currently, rocket fuel emissions have been deemed insignificant when it comes to their consequences to Earth’s environment and LEO. However, emissions will increase in the coming years, making rocket fuel’s contribution to global warming much more significant.

Beyond LEO

Space sustainability concepts and mindsets tend to stay in Low Earth Orbit (LEO). One reason that cannot be ignored is that it is easier to discuss the problem at hand than to speculate on the unknown. There are also examples to prove that since Apollo 17 completed its mission and stayed in Low Earth orbit in 1972, human-crewed space missions in Low Earth orbit have ceased to exist. In this way, it is a reasonable assumption that the closer Moon could be the next object to be explored when the gaze is not limited to LEO. Both lunar orbit and LEO are part of the space environment. In the context of the presence of space debris in LEO, it is normal to speculate that lunar orbit also possesses the nuisance of debris. Space debris measures similar to those in LEO related to space sustainability would be taken.

Not only has the Moon been the subject of study, but other bodies have also been taken into account. Elon Musk, the chief executive of SpaceX at the International Astronautical Congress in 2016, explained the ambitious goal of exploring Mars in the 22nd century. But complicated issues remain, such as the technical aspects of achieving long-distance space flight and the rules and legal aspects associated with the technology, all of which need to be considered.

Calculus

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Calculus

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus and integral calculus; differential calculus concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.

In addition to the differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories which seek to model a particular concept in terms of mathematics. Examples of this convention include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.

History

Main article: History of calculus

Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India.

Ancient precursors

Egypt

Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulae are simple instructions, with no indication as to how they were obtained.

Greece

See also: Greek mathematics

 

Archimedes used the method of exhaustion to calculate the area under a parabola.

Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus (c. 390 – 337 BCE) developed the method of exhaustion to prove the formulas for cone and pyramid volumes.

During the Hellenistic period, this method was further developed by Archimedes, who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus. These problems include, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines.

China

The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere.

Medieval

Middle East

Alhazen, 11th-century Arab mathematician and physicist

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 CE) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.

India

In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".

Modern

Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.

A significant work was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.

The product rule and chain rule, the notions of higher derivatives and Taylor series, and of analytic functions were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.

Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.

 

Isaac Newton developed the use of calculus in his laws of motion and gravitation.

These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.

Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.

When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his Method of Fluxions), but Leibniz published his "Nova Methodus pro Maximis et Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions", a term that endured in English schools into the 19th century. The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi.

Maria Gaetana Agnesi

Foundations

In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.

Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities.^[35] The foundations of differential and integral calculus had been laid. In Cauchy's Cours d'Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can actually validate nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to the complex plane with the development of complex analysis.

In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory, based on earlier developments by Émile Borel, and used it to define integrals of all but the most pathological functions. Laurent Schwartz introduced distributions, which can be used to take the derivative of any function whatsoever.

Limits are not the only rigorous approach to the foundation of calculus. Another way is to use Abraham Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus. There is also smooth infinitesimal analysis, which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of F. W. Lawvere and employing the methods of category theory, smooth infinitesimal analysis views all functions as being continuous and incapable of being expressed in terms of discrete entities. One aspect of this formulation is that the law of excluded middle does not hold. The law of excluded middle is also rejected in constructive mathematics, a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of constructive analysis.

Significance

While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. The Hungarian polymath John von Neumann wrote of this work,

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series.

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, that resolve the paradoxes.

Principles

Limits and infinitesimals

Main articles: Limit of a function and Infinitesimal

Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols ${\displaystyle dx}$ and ${\displaystyle dy}$ were taken to be infinitesimal, and the derivative ${\displaystyle dy/dx}$ was simply their ratio.

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to limits. Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the real number system (as a metric space with the least-upper-bound property). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.

Differential calculus

Main article: Differential calculus

 

Tangent line at (x₀, f(x₀)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating the squaring function turns out to be the doubling function.

In more explicit terms the "doubling function" may be denoted by g(x) = 2x and the "squaring function" by f(x) = x². The "derivative" now takes the function f(x), defined by the expression "x²", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function g(x) = 2x, as will turn out.

In Lagrange's notation, the symbol for a derivative is an apostrophe-like mark called a prime. Thus, the derivative of a function called f is denoted by f′, pronounced "f prime" or "f dash". For instance, if f(x) = x² is the squaring function, then f′(x) = 2x is its derivative (the doubling function g from above).

If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.

If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-intercept, and:

${\displaystyle m={\frac {\text{rise}}{\text{run}}}={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}}.}$

This gives an exact value for the slope of a straight line.^[ If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore, (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is

${\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.}$

This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

${\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.}$

Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.

Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x² be the squaring function.

The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x³ − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.

${\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-3^{2} \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6\end{aligned}}}$

The slope of the tangent line to the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function or just the derivative of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function.

Leibniz notation

Main article: Leibniz's notation

A common notation, introduced by Leibniz, for the derivative in the example above is

${\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}}$

In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:

${\displaystyle {\frac {d}{dx}}(x^{2})=2x.}$

In this usage, the dx in the denominator is read as "with respect to x". Another example of correct notation could be:

${\displaystyle {\begin{aligned}g(t)&=t^{2}+2t+4\\{d \over dt}g(t)&=2t+2\end{aligned}}}$

Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.

Integral calculus

Main article: Integral

 

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).

 

A sequence of Riemann sums over a regular partition of an interval: the total area of the rectangles converges to the integral of the function.

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum.

A motivating example is the distance traveled in a given time. If the speed is constant, only multiplication is needed:

${\displaystyle \mathrm {Distance} =\mathrm {Speed} \cdot \mathrm {Time} }$

But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.

When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with height equal to the velocity and width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve. This connection between the area under a curve and distance traveled can be extended to any irregularly shaped region exhibiting a fluctuating velocity over a given time period. If f(x) represents speed as it varies over time, the distance traveled between the times represented by a and b is the area of the region between f(x) and the x-axis, between x = a and x = b.

To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) traveled in that segment. Associated with each segment is the average value of the function above it, f(x) = h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero.

The symbol of integration is ${\displaystyle \int }$, an elongated S chosen to suggest summation. The definite integral is written as:

${\displaystyle \int _{a}^{b}f(x)\,dx.}$

and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx.

The indefinite integral, or antiderivative, is written:

${\displaystyle \int f(x)\,dx.}$

Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:

${\displaystyle \int 2x\,dx=x^{2}+C.}$

The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.

Fundamental theorem

Main article: Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.

The fundamental theorem of calculus states: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval (a, b), then

${\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).}$

Furthermore, for every x in the interval (a, b),

${\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).}$

This realization, made by both Newton and Leibniz, was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work of Isaac Barrow, is difficult to determine thanks to the priority dispute between them.) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Applications

The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or, it can be used in probability theory to determine the expectation value of a continuous random variable given a probability density function. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points. Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.

Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion, which states that the derivative of an object's momentum with respect to time equals the net force upon it. Alternatively, Newton's second law can be expressed by saying that the net force is equal to the object's mass times its acceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.

Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and in studying radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.

Green's theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter, which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.

In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a cancerous tumour grows.

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.